4-Image Enhancement in the Frequency Domain
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Transcript of 4-Image Enhancement in the Frequency Domain
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Chapter4 Image Enhancement in theFrequency Domain
Enhance: To make greater (as in value, desirability, orattractiveness.
Frequency: The number of times that a periodic
function repeats the same sequence of values during
a unit variation of the independent variable.Websters New Collegiate Dictionary
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Content
Background
Introduction to the Fourier Transform and the
Frequency Domain Smoothing Frequency-Domain Filters
Sharpening Frequency-Domain Filters
Homomorphic Filtering Implementation
Summary
This chapter is concerned primarily with helping the reader develop a
basic understanding of the Fourier transform and the frequency
domain, and how they apply to image enhancement.
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4.1 Background
Any function that periodically repeats itself can be
expressed as the sum of sines and/or cosines of differentfrequencies, each multiplied by a different coefficient
(Fourier series).
Even functions that are not periodic (but whose area
under the curve is finite) can be expressed as the integralof sines and/or cosines multiplied by a weighting function
(Fourier transform).
The advent of digital computation and the discovery of
fast Fourier Transform (FFT) algorithm in the late 1950srevolutionized the field of signal processing, and allowed
for the first time practical processing and meaningful
interpretation of a host of signals of exceptional human
and industrial importance.
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The frequency domain
refers to the plane ofthe two dimensional
discrete Fourier
transform of an image.
The purpose of theFourier transform is to
represent a signal as a
linear combination of
sinusoidal signals of
various frequencies.
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4.2 Introduction to the Fourier Transform
and the Frequency Domain The one-dimensional Fourier transform and its inverse
Fourier transform (continuous case)
Inverse Fourier transform:
The two-dimensional Fourier transform and its inverse
Fourier transform (continuous case)
Inverse Fourier transform:
dueuFxfuxj 2)()(
dydxeyxfvuF
vyuxj )(2),(),(
1where)()( 2
jdxexfuF uxj
dvduevuFyxf vyuxj )(2),(),(
sincos je j
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4.2.1The one-dimensional Fourier transform and
its inverse (discrete time case)
Fourier transform (DFT)
Inverse Fourier transform (IDFT)
The 1/Mmultiplier in front of the Fourier transform sometimes
is placed in the front of the inverse instead. Other times bothequations are multiplied by
Unlike continuous case, the discrete Fourier transform and its
inverse always exist, only iff(x) is finite duration.
1,...,2,1,0for)(1
)(1
0
/2
MuexfM
uFM
x
Muxj
1,...,2,1,0for)()(1
0
/2
MxeuFxfM
u
Muxj
1/ M
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Since and the fact
then discrete Fourier transform can be redefined
Frequency (time) domain: the domain (values ofu) over
which the values ofF(u) range; because u determines
the frequency of the components of the transform.
Frequency (time) component: each of the Mterms of
F(u).
1
0
1( ) ( )[cos 2 / sin 2 / ]
for 0,1, 2,..., 1
M
x
F u f x ux M j ux MM
u M
sincos je j cos)cos(
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F(u) can be expressed in polar coordinates:
R(u): the real part ofF(u)
I(u): the imaginary part ofF(u)
Power spectrum:
)()()()( 222
uIuRuFuP
( )
1/22 2
1
( ) ( )
where ( ) ( ) ( ) (magnitude or spectrum)
( )( ) tan (phase angle or phase spectrum)( )
j uF u F u e
F u R u I u
I uu R u
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Some One-Dimensional Fourier Transform Examples
Please note the relationship between the value of K and the height of
the spectrum and the number of zeros in the frequency domain.
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The transform of a constant function is a DC value only.
The transform of a delta function is a constant.
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The transform of an infinite train of delta functions
spaced by T is an infinite train of delta functions spacedby 1/T.
The transform of a cosine function is a positive delta at
the appropriate positive and negative frequency.
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The transform of a sin function is a negative complex
delta function at the appropriate positive frequency and a
negative complex delta at the appropriate negative
frequency.
The transform of a square pulse is a sinc function.
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4.2.2 The two-dimensional Fourier transform and
its inverse (discrete time case)
Fourier transform (DFT)
Inverse Fourier transform (IDFT)
u, v: the transform or frequency variables
x, y: the spatial or image variables
1,...,2,1,0,1,...,2,1,0for
),(1
),(1
0
1
0
)//(2
NvMu
eyxfMN
vuFM
x
N
y
NvyMuxj
1,...,2,1,0,1,...,2,1,0for
),(),(1
0
1
0
)//(2
NyMx
evuFyxfM
u
N
v
NvyMuxj
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We define the Fourier spectrum, phase anble, and power
spectrum of the two-dimensional Fourier transform as
follows:
R(u,v): the real part ofF(u,v) I(u,v): the imaginary part ofF(u,v)
spectrum)(power),(),(),()(
angle)(phase),(
),(
tan),(
spectrum)(),(),(),(
222
1
2
122
vuIvuRvuFu,vP
vuR
vuI
vu
vuIvuRvuF
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Some properties of Fourier transform:
)(symmetric),(),(
symmetric)(conujgate),(*),(
(average)),(
1
)0,0(
(shift))2
,2
()1)(,(
1
0
1
0
vuFvuF
vuFvuF
yxfMNF
Nv
MuFyxf
M
x
N
y
yx
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(a) f(x,y) (b) F(u,y) (c) F(u,v)
The 2D DFT F(u,v) can be obtained by
1. taking the 1D DFT of every row of image f(x,y), F(u,y),
2. taking the 1D DFT of every column ofF(u,y)
Steps and some example of two-dimensional DFT
y or v
x or u
Convention of coordination:
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shift
Consider the relationship between the separation of zeros in u- or v- direction
and the width or height of white block of source image.
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Shape of three dimensional spectrum
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DFT
DFT
The Property of Two-Dimensional DFT Rotation
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DFT
DFT
DFT
A
B
0.25 * A+ 0.75 * B
The Property of Two-Dimensional DFT Linear Combination
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DFT
DFT
A
Expanding the original image by a factor of n (n=2), filling
the empty new values with zeros, results in the same DFT.
B
The Property of Two-Dimensional DFT Expansion
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Two-Dimensional DFT with Different Functions
Sine wave
Rectangle
Its DFT
Its DFT
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Two-Dimensional DFT with Different Functions
2D Gaussian
function
Impulses
Its DFT
Its DFT
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4.2.3 Filtering in the Frequency DomainFrequency is directly related to rate of change. The frequency of fast
varying components in an image is higher than slowly varying components.
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Basics of Filtering in the Frequency Domain
Including multiplication the input/output image by (-1)x+y.
What is zero-phase-shift filter?
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Some Basic Filters and Their Functions
Multiply all values ofF(u,v) by the filter function (notch filter):
All this filter would do is set F(0,0) to zero (force the average value of
an image to zero) and leave all other frequency components of the
Fourier transform untouched and make prominent edges stand out
otherwise.1
)2/,2/(),(if0),(
NMvuvuH
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Some Basic Filters and Their Functions
Lowpass filter
Highpass filter
Circular symmetry
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Some Basic Filters and Their Functions
Low frequency filters: eliminate the gray-level detail and keep the generalgray-level appearance. (blurring the image)
Low frequency filters: have less gray-level variations in smooth areas and
emphasized transitional (e.g., edge and noise) gray-level detail. (sharpening
images)
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4.2.4 Correspondence between Filtering in the
Spatial and Frequency Domain
Convolution theorem: The discrete convolution of two functions f(x,y) and h(x,y) of
size MNis defined as
The process of implementation:
1) Flipping one function about the origin;
2) Shifting that function with respect to the other by changing the
values of (x, y);
3) Computing a sum of products over all values ofm and n, for
each displacement.
1 1
0 01 1
0 0
1( , ) ( , ) ( , ) ( , )
1( , ) ( , )
M N
m nM N
m n
f x y h x y f m n h x m y nMN
h m n f x m y nMN
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),(),(),(),( vuHvuFyxhyxf
),(),(),(),( vuHvuFyxhyxf Eq. (4.2-32)
Let F(u,v) and H(u,v) denote the Fourier transforms off(x,y) and
h(x,y), then
Eq. (4.2-31)
an impulse function of strengthA, located at coordinates
(x0,y0): and is defined by
:
where : a unit impulse located at the origin
The Fourier transform ofa unit impulse at the origin (Eq4.2-35) :
1
0
1
0
0000 ),(),(),(M
x
N
y
yxAsyyxxAyxs
),( 00 yyxxA
1
0
1
0
)0,0(),(),(M
x
N
y
syxyxs
1
0
1
0
)//(2 1),(1
),(M
x
N
y
NvyMuxj
MNeyx
MNvuF
),( yx
The shifting property of impulse
function
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Let , then the convolution (Eq. (4.2-36))
Combine Eqs. (4.2-35) (4.2-36) with Eq. (4.2-31), weobtain:
),(1
),(),(1),(),(1
0
1
0
yxhMN
nymxhnmMN
yxhyxfM
m
N
n
),(),( yxyxf
),(),(
),(
1
),(
1
),(),(),(),(
),(),(),(),(
vuHyxh
vuHMNyxhMN
vuHyxyxhyx
vuHvuFyxhyxf
That is to say, the response of
impulse input is the transfer
function of filter.
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The distinction and links between spatial and frequency filtering
If the size of spatial and frequency filters is same, then thecomputation burden in spatial domain is larger than in frequency
domain;
However, whenever possible, it makes more sense to filter in the
spatial domain using small filter masks.
Filtering in frequency is more intuitive. We can specify filters in the
frequency, take their inverse transform, and the use the resulting
filter in spatial domain as a guide for constructing smaller spatial
filter masks.
Fourier transform and its inverse are linear process, so thefollowing discussion is limited to linear processes.
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Let H(u) denote a frequency domain, Gaussian filter
function given the equation
where : the standard deviation of the Gaussian curve.
The corresponding filter in the spatial domain is
222
22)( xAexh
22 2/)( uAeuH
There is two reasons that filters based on Gaussian functions are of
particular importance: 1) their shapes are easily specified; 2) both
the forward and inverse Fourier transforms of a Gaussian are real
Gaussian function.
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2 2 2 21 2/ 2 /2
1 2( ) , ( , )u uH u Ae Be A B
The corresponding filter in the spatial domain is
2 2 2 2 2 21 22 2
1 2( ) 2 2x xh x Ae Be
We can note that the value of this types of filter has both negative and
positive values. Once the values turn negative, they never turn positive again.
Filtering in frequency domain is usually used for the guides to design the
filter masks in the spatial domain.
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One very useful property of the Gaussian function is that both it and
its Fourier transform are real valued; there are no complex valuesassociated with them.
In addition, the values are always positive. So, if we convolve animage with a Gaussian function, there will never be any negativeoutput values to deal with.
There is also an important relationship between the widths of aGaussian function and its Fourier transform. If we make the widthof the function smaller, the width of the Fourier transform gets larger.This is controlled by the variance parameter2 in the equations.
These properties make the Gaussian filter very useful forlowpassfiltering an image. The amount of blur is controlled by 2. It can be
implemented in either the spatial or frequency domain. Other filters besides lowpass can also be implemented by using two
different sized Gaussian functions.
Some important properties of Gaussian filters funtions
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4.2 Smoothing Frequency-Domain Filters
The basic model for filtering in the frequency domain
where F(u,v): the Fourier transform of the image to besmoothed
H(u,v): a filter transfer function
Smoothing is fundamentally a lowpass operation in thefrequency domain.
There are several standard forms of lowpass filters (LPF).
Ideal lowpass filter Butterworth lowpass filter
Gaussian lowpass filter
),(),(),( vuFvuHvuG
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Ideal Lowpass Filters (ILPFs)
The simplest lowpass filter is a filter that cuts off all
high-frequency components of the Fourier transform that
are at a distance greater than a specified distance D0
from the origin of the transform.
The transfer function of an ideal lowpass filter
where D(u,v) : the distance from point (u,v) to the center
of ther frequency rectangle (M/2, N/2)
21
22 )2/()2/(),( NvMuvuD
),(if0
),(if1),(
0
0
DvuD
DvuDvuH
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cutoff frequency
ILPF is a type of nonphysical filters and cant be realized with electronic
components and is not very practical.
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The blurring and ringing
phenomena can be seen, inwhich ringing behavior is
characteristic of ideal filters.
A th l f
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Another example ofILPF
Figure 4.13 (a) A frequency-domain
ILPF of radius 5. (b) Corresponding
spatial filter. (c) Five impulses in thespatial domain, simulating the values
of five pixels. (d) Convolution of (b)
and (c) in the spatial domain.frequency
spatial
spatial
spatial
),(),(),(),( vuHvuFyxhyxf
Notation: the radius of center
component and the number of
circles per unit distance from
the origin are inversely
proportional to the value of thecutoff frequency.
diagonal scan line of (d)
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nDvuDvuH
2
0/),(1
1),(
Butterworth Lowpass Filters (BLPFs) with ordern
Note the relationshipbetween ordern and
smoothing
The BLPF may be viewed as a transition between ILPF AND GLPF, BLPF of
order 2 is a good compromise between effective lowpass filtering and
acceptable ringing characteristics.
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Butterworth LowpassFilters (BLPFs)
n=2
D0=5,15,30,80,and 230
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Butterworth Lowpass Filters (BLPFs)Spatial Representation
n=1 n=2 n=5 n=20
l ( )
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Gaussian Lowpass Filters (FLPFs)
20
2 2/),(),( DvuDevuH
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Gaussian LowpassFilters (FLPFs)
D0=5,15,30,80,and 230
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Additional Examples of Lowpass Filtering
Character recognition in machine perception: join the broken character
segments with a Gaussian lowpass filter with D0=80.
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Application in cosmetic processing and produce a smoother,
softer-looking result from a sharp original.
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Gaussian lowpass filter for reducing the horizontal sensor scan lines and
simplifying the detection of features like the interface boundaries.
4 4 Sh i F D i Fil
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( , ) 1 ( , )hp lpH u v H u v
Ideal highpass filter
Butterworth highpass filter
Gaussian highpass filter
),(if1
),(if0),(
0
0
DvuD
DvuDvuH
nvuDDvuH
2
0 ),(/1
1),(
20
2 2/),(1),( DvuDevuH
4.4 Sharpening Frequency Domain Filter
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Highpass Filters Spatial Representations
Id l Hi h Filt (IHPF )
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Ideal Highpass Filters (IHPFs)
),(if1
),(if0),(
0
0
DvuD
DvuDvuH
non-physically realizable with electronic component and have the
same ringing properties as ILPFs.
Butterworth Highpass Filters
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Butterworth Highpass Filters
nvuDDvuH 2
0 ),(/11),(
The result is smoother than that of IHPFs and sharper than that of GHPFs
h l
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Gaussian Highpass Filters
2
0
2
2/),(1),( DvuDevuH
The result is the smoothest in three types of high-pass filters
The Laplacian in the Frequency Domain
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The Laplacian in the Frequency Domain
)(),( 22 vuvuH
2 2
( , ) ( / 2) ( / 2)H u v u M v N
The FT of n-order differential of a function f(x) is
( ) / ( ) ( )n n nd f x dx ju F u
2 22 2 2 2 2
2 2
( , ) ( , )[ ( , )] [ ] ( ) ( , ) ( ) ( , ) ( ) ( , )
f x y f x yf x y ju F u v jv F u v u v F u v
x y
For a two-dimensional function f(x,y), it can be shown that
So, Laplacian can be implemented in the frequency domain by using the filter
Shift the center to (M/2, N/2) and obtain
We have the following Fourier transform pairs
2 2 2( , ) ( / 2) ( / 2) ( , )f x y u M v N F u v
The plot of Laplacian in frequency and spatial domain
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Frequency
domain
Spatial domain
The plot of Laplacian in frequency and spatial domain
2 1 2 2( ) ( ) ( ) 1 (( ) ( ) ) ( )M N
g x y f x y f x y u v F u v
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( , ) ( , ) ( , ) 1 (( ) ( ) ) ( , )2 2
g x y f x y f x y u v F u v
For display
purposes only
A integrated operation
in frequency domain
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Unsharp masking, high-boost filtering, and high-frequency emphasis
filtering (refers to page187-191)
4 5 Homomorphic filtering
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4.5 Homomorphic filteringProblems:
When the illumination radiating to an object is non-
uniform, the detail of the dark part in the image ismore discernable.
aims:Simultaneously compress the gray-level range and
enhance contrast, eliminate the effect of non-uniform
illumination, and emphasis the details.Principal:
Generally, the illumination component of an image is
characterized by slow spatial variations, while the
reflectance components tends to vary abruptly,
particularly at the junctions of dissimilar objects.These characteristics lead to associating the low
frequencies of the Fourier transform of the logarithm
of an image with illumination and the high
frequencies with reflectance.
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),(),(),( yxryxiyxf
The illumination-reflectance model of an image
),( yxi ),( yxrIllumination coefficient: reflectance coefficient:
Steps: ),(ln),(ln),(ln),( yxryxiyxfyxz
)],([ln)],([ln)],([ yxryxiyxz FFF ),(),(),( vuRvuIvuZ
),(),(),(),(),( vuRvuHvuIvuHvuS
2)
1)
3) Determine the H(u, v), which must compress the dynamic
range ofi(x,y), and enhance the contrast ofr(x,y) component.
Th f ll i f ti t th b i
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rL1
2 20( ( , ) / )( , ) ( )[1 ]c D u v DH L LH u v e
The following function meet the above requires
The curve shape shown in above figure can be approximated using basic formof the ideal highpass filters, for example, using a slightly modified form of the
Gaussian highpass filter and can obtain
Steps: 4)
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)],(),([),( 1' vuIvuHyxi F
)],(),([),( 1' vuRvuHyxr F
)],(exp[),(
'
0 yxiyxi
)],(exp[),( '0 yxryxr
),(),(),( 00 yxryxiyxg
Steps: 4)
5)
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ln FFT H(u,v)
FFT-1 exp
f(x,y)
g(x,y)
The flow-chart ofHomomorphic filtering
T l
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Two examples
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4.6 Implementation
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4.6 Implementation4.6.1Some Additional Properties of the 2D Fourier Transform
, distributivity, scaling, and :
0 0
0 0
2 ( / / )0 0
2 ( / / )
0 0
( , ) ( , )
( , ) ( , )
j ux M vy N
j xu M yu N
f x x y y F u v e
f x y e F u u v v
1 2 1 2[ ( , ) ( , )] [ ( , )] [ ( , )]1
( , ) ( / , / )
f x y f x y f x y f x y
f ax by F u a v bab
0 0
( , ) ( , )f r F w
Translation
Distributivity andscaling
rotation
Periodicity, conjugate symmetry, and back-to-back properties
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y j g y y
shift
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
( , ) ( , )
F u v F u M v F u v N F u M v N
f u v f u M v f u v N f u M v N
F u v F u v
Separability
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Separability
1 1 12 / 2 / 2 /
0 0 0
1 1 1( , ) ( , ) ( , )
M N Mj ux M j vy N j ux M
x y x
F u v e f x y e F x v eM N M
A similar process can be applied to computing the 2-D inverse Fourier
transform.
4 6 2 computing the inverse FF using a forward
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4.6.2 computing the inverse FF using a forward
transform algorithmRepeat the one-dimensional inverse FF:
1 2 /
0
( ) ( )M j ux M
u
f x F u e
Take the complex conjugate of two side and multiply M
12 /
0
1 1( ) ( )
Mj ux M
u
f x F u eM M
Which is the form of forward FF. Take the complex conjugate of the result of
the above equation and will get the inverse FF by forward transform.
For two-dimensional case, similarly have
1 12 ( / / )
0 0
1 1( , ) ( , )M N
j ux M vy N
u v
f x y F u v eMN MN
Here, we can treat F(u,v) as a simple function presenting on the forward
transform equation
4.6.3 More on Periodicity: the need for padding
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y p g
1
0
( ) ( )
1( ) ( )
M
m
f x h x
f m h x mM
Convolution process
Aliasing or wraparound error
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extend
extend
The methods of solving the
aliasing problem is to
extend and pad.
( ) 0 1( )
0 1
( ) 0 1
( ) 0 1
where 1
e
e
f x x Af x
A x P
h x x B
h x B x P
P A+ B
For two-dimensional case
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An example
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*
=
4 6 4 convolution and correlation theorems
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4.6.4 convolution and correlation theorems
1 1
0 0
1( , ) ( , ) ( , ) ( , )
M N
m n
f x y h x y f m n h x m y n
MN
1 1
0 0
1( , ) ( , ) ( , ) ( , )
M N
m n
f x y h x y f m n h x m y nMN
Except for the complex offand h not mirrored about the origin, everythingelse in the implementation of correlation is identical to convolution,
including the need for padding.
convolution
correlation
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
f x y h x y F u v H u v
f x y h x y F u v H u v
Correlation theorem
Correlation includes across- and auto-correlation, and its main use is for
matching and sure the location where h (template) finds a correspondence
in f.
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4.6.5 Summary of Some Important Properties
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of the 2-D Fourier Transform
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Summary
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y
The basic concept about Fourier Transform (FT)
The algorithm of FT and the process of frequency filtering
The physical meaning related to FT
The relationship of resolution between spatial and frequency
domain
Correspondence between filtering in the spatial and frequencydomains
The general type of smoothing and sharpening filters and their
main features
Homomorphic filtering Some important properties of FT
Convolution and correlation theorems
Spatial and frequency filtering are both highly subjective processes